In our model, an intense ion current generator with step-like behavior represents the membrane, and a less intense negative current generator (drain) represents AIS. We return to the case we describe in connection with PSP, see Eq. (3.5), with a crucial difference. The flow-in and the flow-out points are at a distance and a “slow” ion current must flow between them. If the current travels to a fixed distance with a fixed speed, as we discuss in connection with Equ. (2.2), we expect that the output current appears with a delay compared to the input current. We assume that the charge conserves, i.e., the input current equals the output current. That means, for a one-dimensional membrane, we shall write Kirchoff’s Junction Law in the form
| (3.9) |
instead of the usual form, without delay. In Fig. 1 in [50] one can see that the current travels with speed less than 1 cm/s toward AIS, and that Kirchoff’s Law is valid only in the form given by Eq. (3.9). It is, essentially, what the telegraph equation expresses for technical computing: the macroscopic current has finite speed.
We assume that the input current due to the rushed-in ions is similar to the one we derived in connection with PSP see Equ. (3.5). That is, we expect that the resulting (net) current is a “ghost” image shown in Fig. 3.23, which can be interpreted as a kind of interference (a difference between a positive current and shifted negative current) between the input and output currents, and expresses Kirchoff’s Junction Law for “slow” current in a neuron. For the figure, we assumed , and the parameters used to generate the function displayed in Fig. 3.25. The negative output current has been observed and measured by [9], see their Fig. 18, but – due to the lack of the idea of “slow current”, furthermore using the mistakenly measured empirical dependencies of “conductivity” – it has been identified as outward current. A high sodium channel density is present in the AIS [50] to form AP, and imaging ions show (see their Fig. 3f in [50]) that ions arrive at it. Hypothesizing in all cases leads to some discrepancy see for example, ’It is counter- intuitive that removing a potassium conductance would decrease the excitability of a neuron’ [117], and
In our model, the membrane acts as a voltage generator with the time course described by Eq. (3.6), with the appropriate coefficients. This change is a drastic departure from the classic picture using a current generator, where a “fast” current flows through the resistor, generating a voltage that charges the capacitor. Initially, the capacitor is empty, so it will temporarily store the charge; that charge produces the ’damping’ contribution later but cannot explain a negative contribution. To explain the experienced hyperpolarization, a current in the opposite direction must be assumed, although it has no source charge in the membrane and no experimental proof underpins its existence during evoking an AP; see [117].
In our modern picture, the initial ion inflow saturates, and the relatively low-intensity slow current removes the charges from the membrane’s surface. The membrane attempts to remain equipotential despite the experienced current drain, but the slow current needs time to reach the AIS. The interplay of the finite-speed current flowing on the finite-size surface and the voltage-dependent exponential outflow shape the AP. There is no need to assume the inflow or outflow of specific currents and the change of ion type. The extended size of the membrane, accompanied by slow ion propagation, entirely explains why the spikes are issued and provides its parameters. Similarly, no control mechanism is needed: biology takes advantage of the slow ion propagation speed. (The function displayed should be convolved with a function considering the distribution of distances between the input and output points; i.e., consider an actual membrane shape).
To describe how the neuron’s membrane forms an AP, we consider that the membrane becomes highly charged (i.e., will have a considerable potential) after opening its ion channels. That potential difference will drive a macroscopic current toward the AIS, where a macroscopic current flows out, as described in [50]. The mathematical formalism is the same as in the case of PSP, see Eq. (3.5), except that the current inflow is more intense, given that the membrane’s surface is much larger. Although the AIS is much smaller, its much higher ion channel density [50] enables it to forward that intense ”longitudinal” current toward the axon (where it is transmitted as a ”transversal current”; see good textbooks and our discussion). As we discussed, an AP can be described by three parameters: how the rush-in current rises (a function of the area and ion channel density), how the rushed-in charge can flow out (including how long the current path is), and the parameters of the neuronal circuit.
That current on the condenser with capacity , alone, would produce a voltage change : it is the input side of the circuit. The membrane (in cooperation with the AIS) behaves as a differentiator circuit. It will significantly change the form of the voltage’s time course on its output side (as discussed in section 2.2.4, one can imitate the effects of a “slow” current flowing on a system with distributed parameters using equations created for discrete parameter case). The membrane potential produces a current that discharges the condenser, decreasing the potential generated by the membrane’s current.
Our model hypothesizes that the current due to the rushed-in ions maintains the time course of the voltage derivative in the input side, see Eq.(3.7). We shall solve the equation numerically to receive the output voltage, the AP. The shape of the voltage due to the slow current on the membrane, described by Eq. (3.5) and its derivative, described by Eq. (3.6), are depicted in Fig. 3.11; with the parameters concluded from Fig. 3.22. The formalism and model are the same also in the case of a membrane; only the coefficients are different.
By varying those parameters, a variety of AP shapes can be described using the same model, see Fig. 3.10. The various colors and line types demonstrate the influence of parameter values on the calculated shape of the AP. We based our calculations on a resting potential of -65 mV and a threshold offset potential of +20 mV. The red lines represent AP for current intensities similar to those used to generate the diagram line in Fig. 3.25. The green and blue lines depict the AP for higher and lower intensity currents, respectively. The continuous lines show the AP for a neuronal oscillator with capacity we used to generate Fig. 3.25. The dotted and dashed lines represent circuits with higher and lower time constant values, respectively. Our research suggests that assuming a differentiator-type circuit for the neuronal membrane can imitate the effects of the “slow” current’s temporal behavior, see Fig. 3.23. As we discussed in section 2.2.4, the time constant drastically influences the resemblance of the (PSP-like) input function shape and the (AP-like) output shape. Furthermore, the higher the charge-up current compared to the fixed-value output current (defined by ), the more resemblant the output voltage shape and the empirical AP.
In the picture, we suggest here, the voltage of the membrane increases enormously (described in a physically plausible way and with a mathematically and physically correct time-dependent function), observed as large transient local voltages [80, 81]. The membrane, acting as a semipermeable insulator surface, hosts charge carriers that distribute on it, capable of reaching other areas with finite surface speed. Consequently, the AIS will experience only a marginal increase upon receiving an axonal input, and only with a delay. A temporally distributed charge packet is the sole factor that evokes the observed voltage increase, without any other assumed in- and outflow of ions. This novel approach to understanding the initiation of APs sets our model apart.